OPTIMAL CONTROL AND STABILIZATION FOR DISCRETE-TIME MARKOV JUMP LINEAR SYSTEMS WITH INPUT DELAY

This paper examines the finite horizon optimal control problem and infinite horizon stabilization problem for Markov jump linear system (MJLS) with input delay. The essential obstacle encountered in this paper is that the optimal controller cannot be represented as a product of a deterministic gain and a state predictor. This paper presents a complete solution to the problems: (1) For the finite horizon, the necessary and sufficient solvability condition of the optimal control and the analytical controller are given based on the modified coupled difference Riccati equation defined herein. (2) For the infinite horizon, the necessary and sufficient stabilization conditions are explored under explicit expressions, and the optimal controller is designed with a modified coupled algebraic Riccati equation (CARE). We show that under the exact observability assumption, the MJLS with input delay is stabilizable in the mean-square sense with the optimal controller if and only if the CARE has a particular positive definite solution. The introduction of a delayed forward and backward Markov difference equation and the definition of a new type of Lyapunov function form the basic tools to solving this problem.